text: The Newton's constitutive law for a compressible flow results from the following assumptions on the Cauchy stress tensor:
  Since the trace of the rate-of-strain tensor in three dimensions is the divergence (i.e. rate of expansion) of the flow:
  Given this relation, and since the trace of the identity tensor in three dimensions is three:
  the trace of the stress tensor in three dimensions becomes:
  So by alternatively decomposing the stress tensor into isotropic and deviatoric parts, as usual in fluid dynamics:
  Introducing the bulk viscosity ,
  we arrive to the linear constitutive equation in the form usually employed in thermal hydraulics:
  which can also be arranged in the other usual form:
  Note that in the compressible case the pressure is no more proportional to the isotropic stress term, since there is the additional bulk viscosity term:
  and the deviatoric stress tensor  is still coincident with the shear stress tensor  (i.e. the deviatoric stress in a Newtonian fluid has no normal stress components), and it has a compressibility term in addition to the incompressible case, which is proportional to the shear viscosity:
  Note that the incompressible case correspond to the assumption that the pressure constrains the flow so that the volume of fluid elements is constant: isochoric flow resulting in a solenoidal velocity field with .
  So one returns to the expressions for pressure and deviatoric stress seen in the preceding paragraph. 
  Both bulk viscosity  and dynamic viscosity  need not be constant – in general, they depend on two thermodynamics variables if the fluid contains a single chemical species, say for example, pressure and temperature. Any equation that makes explicit one of these transport coefficient in the conservation variables is called an equation of state.
  Apart from its dependence of pressure and temperature, the second viscosity coefficient also depends on the process, that is to say, the second viscosity coefficient is not just a material property. Example: in the case of a sound wave with a definitive frequency that alternatively compresses and expands a fluid element, the second viscosity coefficient depends on the frequency of the wave. This dependence is called the dispersion. In some cases, the second viscosity  can be assumed to be constant in which case, the effect of the volume viscosity  is that the mechanical pressure is not equivalent to the thermodynamic pressure: as demonstrated below.
  However, this difference is usually neglected most of the time (that is whenever we are not dealing with processes such as sound absorption and attenuation of shock waves, where second viscosity coefficient becomes important) by explicitly assuming . The assumption of setting  is called as the Stokes hypothesis. The validity of Stokes hypothesis can be demonstrated for monoatomic gas both experimentally and from the kinetic theory; for other gases and liquids, Stokes hypothesis is generally incorrect.
  Finally, note that Stokes hypothesis is less restrictive that the one of incompressible flow. In fact, in the incompressible flow both the bulk viscosity term, and the shear viscosity term in the divergence of the flow velocity term disappears, while in the Stokes hypothesis the first term also disappears but the second one still remains.